Spring 2019 Homework Assignments
A dielectric sphere with uniform permittivity ε and radius R is placed at the origin.
Also at the origin is a point charge Q.
(a) Find the polarizarion P (vector!) of the sphere.
(b) The sphere is now rotated with angular velocity ω around the z-axis. Remember
that the H field for a moving material takes the form H = B/μo - M - P x v
where M is the magnetization and v is the velocity of the material.
This choice allows one to write ∇ x H = Jfree. Since Jfree = 0
in our problem, one has ∇ x H = 0 so that one may obtain H from a
scalar potential: H = -∇Φm. The source ("charge") for H is then given by
ρm = ∇⋅H. Find ρm based on the discussion above. Note that ρm may have
a surface contribution as well as a volume term.
(c) Find Φm and H corresponding to ρm.
Remember that the response of the field D to the electric field E is related
through the response function G(t) so that
D(t) = εo + ∫o∞ G(τ) E(t-τ) dτ.
G(t) is related to the complex dielectric function ε(ω) through
ε(ω) / εo -1 = ∫-∞∞ G(t) exp(iωt) dt.
Given that G(t) is of the form
{ 0 for t<0
G(t) = {
{ α sin(ωot) / (exp(γt) - 1) otherwise
where α, ωo and γ are constants,
(a) find the corresponding ε(ω). (Hint: expand G(t) in powers of exp(-γt).)
(b) Does the analytic form of ε(ω) satisfy the condition for causality? How?
Consider two equal and opposite charges +q and -q placed on the z-axis at coordinates ±a/2 respectively. At time t=0 the two charges move towards each other with speed v and annihilate one another. (You can treat the charge density as a vanishing dipole.) What fields are observed at a large distance r from the origin as a function of time? (You will need to keep terms only of order 1/r.) What is the total power radiated into the far-zone?
Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.